Question 32862: Both Values in a Matrix Scalar Identity

Original question and givens

Over R, the 2×2 matrix is M=[[-1, 4], [-2, 12]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[-1, 12] and c=-2, compute T(cu) and cT(u). Answer as left vector = right vector.

• G001 — Over R, the 2×2 matrix is M=[[-1, 4], [-2, 12]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[-1, 12] and c=-2, compute T(cu) and cT(u). Answer as left vector = right vector.

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

• mko-linear-transformation

Step-by-step derivation

source roles = [qf-linear-map-homogeneity-values, homogeneity_values, reals, []]

derived = [[VECTOR_EQUALS_VECTOR_STRING]]

Homogeneity ORIGINAL c times u ORIGINAL component0 := mul(-2, -1)

Homogeneity ORIGINAL c times u ORIGINAL component0 = 2

Homogeneity ORIGINAL c times u ORIGINAL component1 := mul(-2, 12)

Homogeneity ORIGINAL c times u ORIGINAL component1 = -24

Homogeneity LEFT M of scaled input ORIGINAL row0 column0 product := mul(-1, 2)

Homogeneity LEFT M of scaled input ORIGINAL row0 column0 product = -2

Homogeneity LEFT M of scaled input ORIGINAL row0 column1 product := mul(4, -24)

Homogeneity LEFT M of scaled input ORIGINAL row0 column1 product = -96

Homogeneity LEFT M of scaled input ORIGINAL row0 two products sum := sum(-2, -96)

Homogeneity LEFT M of scaled input ORIGINAL row0 two products sum = -98

source roles = [Homogeneity LEFT M of scaled input, 0, [-1, 4], [2, -24]]

derived = [[-2, -96], -98]

Homogeneity LEFT M of scaled input ORIGINAL row1 column0 product := mul(-2, 2)

Homogeneity LEFT M of scaled input ORIGINAL row1 column0 product = -4

Homogeneity LEFT M of scaled input ORIGINAL row1 column1 product := mul(12, -24)

Homogeneity LEFT M of scaled input ORIGINAL row1 column1 product = -288

Homogeneity LEFT M of scaled input ORIGINAL row1 two products sum := sum(-4, -288)

Homogeneity LEFT M of scaled input ORIGINAL row1 two products sum = -292

source roles = [Homogeneity LEFT M of scaled input, 1, [-2, 12], [2, -24]]

derived = [[-4, -288], -292]

Homogeneity RIGHT M of u ORIGINAL row0 column0 product := mul(-1, -1)

Homogeneity RIGHT M of u ORIGINAL row0 column0 product = 1

Homogeneity RIGHT M of u ORIGINAL row0 column1 product := mul(4, 12)

Homogeneity RIGHT M of u ORIGINAL row0 column1 product = 48

Homogeneity RIGHT M of u ORIGINAL row0 two products sum := sum(1, 48)

Homogeneity RIGHT M of u ORIGINAL row0 two products sum = 49

source roles = [Homogeneity RIGHT M of u, 0, [-1, 4], [-1, 12]]

derived = [[1, 48], 49]

Homogeneity RIGHT M of u ORIGINAL row1 column0 product := mul(-2, -1)

Homogeneity RIGHT M of u ORIGINAL row1 column0 product = 2

Homogeneity RIGHT M of u ORIGINAL row1 column1 product := mul(12, 12)

Homogeneity RIGHT M of u ORIGINAL row1 column1 product = 144

Homogeneity RIGHT M of u ORIGINAL row1 two products sum := sum(2, 144)

Homogeneity RIGHT M of u ORIGINAL row1 two products sum = 146

source roles = [Homogeneity RIGHT M of u, 1, [-2, 12], [-1, 12]]

derived = [[2, 144], 146]

Homogeneity RIGHT c times image ORIGINAL component0 := mul(-2, 49)

Homogeneity RIGHT c times image ORIGINAL component0 = -98

Homogeneity RIGHT c times image ORIGINAL component1 := mul(-2, 146)

Homogeneity RIGHT c times image ORIGINAL component1 = -292

source roles = [homogeneity_values, [[-1, 4], [-2, 12]], [-1, 12], -2]

derived = [[-98, -292], [-98, -292], [-98, -292] = [-98, -292]]

Reverse checks

• S022 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
• S021 · premise_and_result_audit — Recompute Homogeneity RIGHT c times image ORIGINAL component1 with exact rational arithmetic, not rounded decimals.
• S020 · premise_and_result_audit — Recompute Homogeneity RIGHT c times image ORIGINAL component0 with exact rational arithmetic, not rounded decimals.
• S019 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
• S018 · premise_and_result_audit — Recompute Homogeneity RIGHT M of u ORIGINAL row1 two products sum with exact rational arithmetic, not rounded decimals.
• S017 · premise_and_result_audit — Recompute Homogeneity RIGHT M of u ORIGINAL row1 column1 product with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Recompute Homogeneity RIGHT M of u ORIGINAL row1 column0 product with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
• S014 · premise_and_result_audit — Recompute Homogeneity RIGHT M of u ORIGINAL row0 two products sum with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute Homogeneity RIGHT M of u ORIGINAL row0 column1 product with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute Homogeneity RIGHT M of u ORIGINAL row0 column0 product with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
• S010 · premise_and_result_audit — Recompute Homogeneity LEFT M of scaled input ORIGINAL row1 two products sum with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute Homogeneity LEFT M of scaled input ORIGINAL row1 column1 product with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute Homogeneity LEFT M of scaled input ORIGINAL row1 column0 product with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
• S006 · premise_and_result_audit — Recompute Homogeneity LEFT M of scaled input ORIGINAL row0 two products sum with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute Homogeneity LEFT M of scaled input ORIGINAL row0 column1 product with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute Homogeneity LEFT M of scaled input ORIGINAL row0 column0 product with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute Homogeneity ORIGINAL c times u ORIGINAL component1 with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute Homogeneity ORIGINAL c times u ORIGINAL component0 with exact rational arithmetic, not rounded decimals.
• S001 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.

Result

[-98, -292] = [-98, -292]

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.